# Four beautiful quadrature rules

## TL;DR

This paper introduces a framework for deriving quadrature rules as weighted means of companion rules, exemplified by the trapezoidal and Simpson rules, and demonstrates their application in approximating pi with numerical examples.

## Contribution

The paper presents a novel framework linking companion quadrature rules and their associate rules, providing new insights into classical numerical integration methods.

## Key findings

- The trapezoidal and Simpson rules are weighted means of specific companion rules.
- The rules reproduce the value of pi through exact integration of a sine-squared function.
- Numerical examples show effective approximation of pi using composite rules.

## Abstract

A framework is presented to compute approximations of an integral $I(f)=\displaystyle \int_a^b f(x) dx$ from a pair of companion rules and its associate rule. We show that an associate rule is a weighted mean of two companion rules. In particular, the trapezoidal (T) and Simpson (S) rules are weighted means of the companion pairs (L,R) and (T,M) respectively, with L the left rectangle, R the right rectangle and M the midpoint rules. As L,R,T and M reproduce exactly the number $\pi=\displaystyle \int_0^\pi 2\, \sin^2(x) dx$, we named them the four "{beautiful}" \ rules. For this example the geometrical interpretation of the rules suggest possible applications of the transcendental number $\pi$ in architectural design, justifying the attribute beautiful given to the mentioned rules. As a complement we consider other appropriate integrand functions $f$, applying composite rules in order to obtain good approximations of $\pi$, as shown in the worked numerical examples.